Module 7 Real World Applications
Conic Sections
EarthquakeSeismologists have long used conic sections to help them locate the epicenter, or source,
of an earthquake. Recording stations throughout the country have the capability to tell
how far away an earthquake is…although they can’t always tell the specific direction.
The stations become the center of a circle. The distance to the epicenter becomes the
radius of the circle.
With only one station, the earthquake location is narrowed down to a radius of points
surrounding the station. When two stations are used, scientists can graph two circles –
one based on information from each station. This will narrow down the possible
epicenter locations to places where the two circles intersect. This might be 1 or 2 points.
To finalize the location of an epicenter, three stations are used. For this data, the
intersection of these three circles will be a single point, and that point is the epicenter.
Problem 1:
An earthquake has occurred. You have information from three
recording stations as to the earthquake’s location. The locations
are given as if a coordinate plane has been laid over a map:
Station #1 -
Located at (0, 0), Earthquake was √40 miles away
Station #2 -
Located at (5, 6), Earthquake was 3 miles away
Station #3 -
Located at (-1, 11), Earthquake was √34 miles away
a)
Write the standard form equations for the circles formed from each station’s data.
b)
Identify the epicenter of the earthquake. (Find the point of intersection of the
three circles.)
Mirrors and LensesIf you take a parabola and rotate it in three-dimensional space, you get a figure called a
paraboloid. A picture of a paraboloid can be found here:
http://en.wikipedia.org/wiki/Paraboloid
You’re probably familiar with the paraboloid in the form of a satellite dishes, flashlights,
headlights, etc. Paraboloids are commonly used to reflect or focus sound, light, radio
waves, etc.
The key is that a signal source at the focus of the paraboloid will reflect off waves in lines
parallel to the axis of symmetry. The reverse is also true, waves coming in parallel into
the paraboloid will meet at the focus.
If you’ve watched any football games recently, you’ve probably seen a parabolic
reflector with a microphone in the middle. This is used to gather sound from the field far
away and focus it for recording.
Example:
A parabolic reflector is 4 feet across and 1 foot deep. Where should the
microphone be placed so that it is at the focus of this paraboloid?
You can draw a cross section of the microphone on a coordinate plane. We put the vertex
at the origin, (0, 0). We know that the parabola is 4 feet across and 1 foot deep, so we
plot the points (-2, 1) and (2, 1). Those must be on our parabola.
Here’s a sketch of the parabola:
We must find the equation for the parabola. We can then find ‘p’.
Since our parabola opens up, we know it has this form:
Let’s plug in our values for ‘x’ and ‘y’ to find p:
x 2  4 py
x 2  4 py
(2) 2  4 p(1)
4  4p
Our value for p = 1. The equation for our parabola is: x 2  4(1) y
1 p
So our microphone should be placed 1 foot above the vertex.
Problem 2:
A flashlight is a paraboloid of revolution. Its diameter is 6 cm and its
depth is 2 cm. How far from the vertex should the filament of the light
bulb be placed for the flashlight to have its beam run parallel to the axis of
its mirror?
Problem 3:
A company produces headlights for cars which are parabolic. If the
equation for the headlights is x 2  20 y , how far from the vertex should
the bulb be placed?
OrbitMany planets and other astronomical bodies have elliptical orbits. One of our most
famous comets, Halley’s Comet, has an elliptical orbit with the sun at one focus.
The length of the major axis of this orbit is approximately 36.2 AU and the minor axis
length is 9.12 AU. AU stands for ‘Astronomical Unit’ and 1 AU is approximately
93,000,000 miles – the distance from the Earth to the Sun.
Problem 4:
Find the eccentricity of the orbit of Halley’s Comet
LithotripterEllipses have a special property because of their shape. In a three-dimensional ellipse, or
ellipsoid of revolution, waves which emanate from one focus reflect off the surface of the
ellipse to the other focus.
You can see a nice picture at: http://en.wikipedia.org/wiki/Ellipsoid
In my picture, I’ve tried to show different routes that waves might take from the focus on
the left to the focus on the right.
One use of this property is in a Whispering Chamber. The Museum of Science and
Industry has a large Whispering Gallery. You stand at one focus and you can have a
conversation with someone 40 feet away at the other focus.
One important use of this property is a lithotripter. This is a device used to shatter
kidney stones using ultrahigh-frequency shock waves. The source of these waves is one
focus of the ellipse. The patient is positioned so that the stones are at the other focus.
Another place an ellipse is seen often is in Kepler’s Second Law of Planetary Motion:
http://en.wikipedia.org/wiki/Kepler's_laws_of_planetary_motion
Problem 5:
Put Kepler’s Second Law into your own words. You might want to
explore the internet for more sources that Wikipedia.
Random Knowledge……
LORANLORAN stands for LOng RAnge Navigation. It’s a type of radar navigation system used
by pilots.
Radio waves are sent out by two stations, A and B. A plane is located at position C. The
plane is at the point of intersection of two hyperbolas. One with foci A and C, the other
with foci B and C.
Using algebraic techniques, LORAN receptors can find the intersection of these
hyperbolas, and then use that information to tell how far away the plane is from each of
these stations. The pilot can then ascertain their location.